Many honing mandrels and other honing devices have been constructed and used in the past. For the most part, the known constructions have used honing stones and other abrasive members which are adjustable radially during a honing operation in order to maintain them engaged with a work surface as the work surface is enlarged and as the stones wear. Typical of such honing mandrels are the mandrels disclosed in Sunnen U.S. Pat. Nos. 2,532,682, dated Dec. 5, 1950; 2,580,327, dated Dec. 25, 1951; 2,580,328; dated Dec. 25, 1951; 2,799,127; dated July 16, 1957; 2,815,615, dated Dec. 10, 1957; and 3,800,482, dated Apr. 2, 1974. Honing devices of the types disclosed in these patents are well known and widely used and the present construction is not designed or constructed to replace them. Another type of work engaging honing device is disclosed in Althen et al U.S. Pat. No. 4,197,680. The honing device disclosed in this copending application is pertinent to the present construction but does not disclose the use of more than one differently axially tapered portion to produce very accurate honing, and this feature of the present construction in addition to producing accurate honed surfaces also provides important advantages over the known prior art as will be explained.